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Numerical Instabilities (numerical + instability)
Selected AbstractsNumerical instabilities in the computation of pseudopotential matrix elementsJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 2 2006Christoph van Wüllen Abstract Steep high angular momentum Gaussian basis functions in the vicinity of a nucleus whose inner electrons are replaced by an effective core potential may lead to numerical instabilities when calculating matrix elements of the core potential. Numerical roundoff errors may be amplified to an extent that spoils any result obtained in such a calculation. Effective core potential matrix elements for a model problem are computed with high numerical accuracy using the standard algorithm used in quantum chemical codes and compared to results of the MOLPRO program. Thus, it is demonstrated how the relative and absolute errors depend an basis function angular momenta, basis function exponents and the distance between the off-center basis function and the center carrying the effective core potential. Then, the problem is analyzed and closed expressions are derived for the expected numerical error in the limit of large basis function exponents. It is briefly discussed how other algorithms would behave in the critical case, and they are found to have problems as well. The numerical stability could be increased a little bit if the type 1 matrix elements were computed without making use of a partial wave expansion. © 2005 Wiley Periodicals, Inc., J Comput Chem 27: 135,141 2006 [source] Dynamic Wavelet Neural Network for Nonlinear Identification of Highrise BuildingsCOMPUTER-AIDED CIVIL AND INFRASTRUCTURE ENGINEERING, Issue 5 2005Xiaomo Jiang Compared with conventional neural networks, training of a dynamic neural network for system identification of large-scale structures is substantially more complicated and time consuming because both input and output of the network are not single valued but involve thousands of time steps. In this article, an adaptive Levenberg,Marquardt least-squares algorithm with a backtracking inexact linear search scheme is presented for training of the dynamic fuzzy WNN model. The approach avoids the second-order differentiation required in the Gauss,Newton algorithm and overcomes the numerical instabilities encountered in the steepest descent algorithm with improved learning convergence rate and high computational efficiency. The model is applied to two highrise moment-resisting building structures, taking into account their geometric nonlinearities. Validation results demonstrate that the new methodology provides an efficient and accurate tool for nonlinear system identification of high-rising buildings. [source] Polygonal finite elements for topology optimization: A unifying paradigmINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2010Cameron Talischi Abstract In topology optimization literature, the parameterization of design is commonly carried out on uniform grids consisting of Lagrangian-type finite elements (e.g. linear quads). These formulations, however, suffer from numerical anomalies such as checkerboard patterns and one-node connections, which has prompted extensive research on these topics. A problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members, leading to mesh-dependent sub-optimal designs. Thus, to address the geometric features of the spatial discretization, we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions. More specifically, we consider polygonal meshes constructed from Voronoi tessellations, which in addition to possessing higher degree of geometric isotropy, allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities. Copyright © 2009 John Wiley & Sons, Ltd. [source] Theoretical aspects of the internal element connectivity parameterization approach for topology optimizationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2008Gil Ho Yoon Abstract The internal element connectivity parameterization (I-ECP) method is an alternative approach to overcome numerical instabilities associated with low-stiffness element states in non-linear problems. In I-ECP, elements are connected by zero-length links while their link stiffness values are varied. Therefore, it is important to interpolate link stiffness properly to obtain stably converging results. The main objective of this work is two-fold (1) the investigation of the relationship between the link stiffness and the stiffness of a domain-discretizing patch by using a discrete model and a homogenized model and (2) the suggestion of link stiffness interpolation functions. The effects of link stiffness penalization on solution convergence are then tested with several numerical examples. The developed homogenized I-ECP model can also be used to physically interpret an intermediate design variable state. Copyright © 2008 John Wiley & Sons, Ltd. [source] Energy transfer in master equation simulations: A new approachINTERNATIONAL JOURNAL OF CHEMICAL KINETICS, Issue 12 2009John R. BarkerArticle first published online: 8 OCT 200 Collisional energy transfer plays a key role in recombination, unimolecular, and chemical activation reactions. For master equation simulations of such reaction systems, it is conventionally assumed that the rate constant for inelastic energy transfer collisions is independent of the excitation energy. However, numerical instabilities and nonphysical results are encountered when normalizing the collision step-size distribution in the sparse density of states regime at low energies. It is argued here that the conventional assumption is not correct, and it is shown that the numerical problems and nonphysical results are eliminated by making a plausible assumption about the energy dependence of the rate coefficient for inelastic collisions. The new assumption produces a model that is more physically realistic for any reasonable choice of collision step-size distribution, but more work remains to be done. The resulting numerical algorithm is stable and noniterative. Testing shows that overall accuracy in master equation simulations is better with this new approach than with the conventional one. This new approach is appropriate for all energy-grained master equation formulations. © 2009 Wiley Periodicals, Inc. Int J Chem Kinet 41: 748,763, 2009 [source] Numerical instabilities in the computation of pseudopotential matrix elementsJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 2 2006Christoph van Wüllen Abstract Steep high angular momentum Gaussian basis functions in the vicinity of a nucleus whose inner electrons are replaced by an effective core potential may lead to numerical instabilities when calculating matrix elements of the core potential. Numerical roundoff errors may be amplified to an extent that spoils any result obtained in such a calculation. Effective core potential matrix elements for a model problem are computed with high numerical accuracy using the standard algorithm used in quantum chemical codes and compared to results of the MOLPRO program. Thus, it is demonstrated how the relative and absolute errors depend an basis function angular momenta, basis function exponents and the distance between the off-center basis function and the center carrying the effective core potential. Then, the problem is analyzed and closed expressions are derived for the expected numerical error in the limit of large basis function exponents. It is briefly discussed how other algorithms would behave in the critical case, and they are found to have problems as well. The numerical stability could be increased a little bit if the type 1 matrix elements were computed without making use of a partial wave expansion. © 2005 Wiley Periodicals, Inc., J Comput Chem 27: 135,141 2006 [source] Non-linear behavior of mass concrete in three-dimensional problems using a smeared crack approachEARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 3 2005H. Mirzabozorg Abstract A smeared crack approach has been proposed to model the static and dynamic behavior of mass concrete in three-dimensional space. The proposed model simulates the tensile fracture on the mass concrete and contains pre-softening behavior, softening initiation, fracture energy conservation and strain rate effects under dynamic loads. The validity of the proposed model has been checked using the available experimental results under static and dynamic loads. The direct and indirect displacement control algorithms have been employed under incremental increasing static loads. It was found that the proposed model gives excellent results and crack profiles when compared with the available data under static loads. The Koyna Dam in India has been used to verify the dynamic behavior of the proposed model. It was found that the resulting crack profiles were in good agreement with the available experimental results. Finally, the Morrow Point Dam was analyzed, including the dam,reservoir interaction effects, to consider its non-linear seismic behavior. It was found that the resulting crack profiles were in good agreement with the contour of maximum principal stresses and no numerical instability occurred during the analysis. Copyright © 2004 John Wiley & Sons, Ltd. [source] On the residue calculus evaluation of the 3-D anisotropic elastic green's functionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 5 2004A.-V. Phan Abstract An algorithm based upon the residue calculus for computing three-dimensional anisotropic elastic Green's function and its derivatives has been presented in Sales and Gray (Comput. Structures 1998; 69:247,254). It has been shown that the algorithm runs three to four times faster than the standard Wilson,Cruse interpolation scheme. However, the main concern of the Sales,Gray algorithm is its numerical instability that could lead to significant errors due to the existence of multiple poles of the residue. This paper proposes a remedy for the problem by adding the capability to evaluate the Green's function in case of multiple poles of the residue. Further, an improved numerical implementation based on the use of double-subscript-notation elastic constants in determining the Christoffel tensor is also at issue. Copyright © 2004 John Wiley & Sons, Ltd. [source] DSC-Ritz method for high-mode frequency analysis of thick shallow shellsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2005C. W. Lim Abstract This paper addresses a challenging problem in computational mechanics,the analysis of thick shallow shells vibrating at high modes. Existing methods encounter significant difficulties for such a problem due to numerical instability. A new numerical approach, DSC-Ritz method, is developed by taking the advantages of both the discrete singular convolution (DSC) wavelet kernels of the Dirichlet type and the Ritz method for the numerical solution of thick shells with all possible combinations of commonly occurred boundary conditions. As wavelets are localized in both frequency and co-ordinate domains, they give rise to numerical schemes with optimal accurate, stability and flexibility. Numerical examples are considered for Mindlin plates and shells with various edge supports. Benchmark solutions are obtained and analyzed in detail. Experimental results validate the convergence, stability, accuracy and reliability of the proposed approach. In particular, with a reasonable number of grid points, the new DSC-Ritz method is capable of producing highly accurate numerical results for high-mode vibration frequencies, which are hitherto unavailable to engineers. Moreover, the capability of predicting high modes endows us the privilege to reveal a discrepancy between natural higher-order vibration modes of a Mindlin plate and those calculated via an analytical relationship linking Kirchhoff and Mindlin plates. Copyright © 2004 John Wiley & Sons, Ltd. [source] Godunov-type adaptive grid model of wave,current interaction at cuspate beachesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2004Benedict D. Rogers Abstract This paper presents a second-order accurate Godunov-type numerical scheme for depth- and period-averaged wave,current interaction. A flux Jacobian is derived for the wave conservation equations and its eigensystem determined, enabling Roe's approximate Riemann solver to be used to evaluate convective fluxes. Dynamically adaptive quadtree grids are used to focus on local hydrodynamic features, where sharp gradients occur in the flow variables. Adaptation criteria based on depth-averaged vorticity, wave-height gradient, wave steepness and the magnitude of velocity gradients are found to produce accurate solutions for nearshore circulation at a half-sinusoidal beach. However, the simultaneous combination of two or more separate criteria produces numerical instability and interference unless all criteria are satisfied for mesh depletion. Simulations of wave,current interaction at a multi-cusped beach match laboratory data from the United Kingdom Coastal Research Facility (UKCRF). A parameter study demonstrates the sensitivity of nearshore flow patterns to changes in relative cusp height, angle of wave incidence, bed roughness, offshore wave height and assumed turbulent eddy viscosity. Only a small deviation from normal wave incidence is required to initiate a meandering longshore current. Nearshore circulation patterns are highly dependent on the offshore wave height. Reduction of the assumed eddy viscosity parameter causes the primary circulation cells for normally incident waves to increase in strength whilst producing rip-like currents cutting diagonally across the surf zone. Copyright © 2004 John Wiley & Sons, Ltd. [source] | |||||||||||||